Finite-dimensional left ideals in some algebras associated with a locally compact group

Abstract:

Let $G$ be a locally compact group, let $L^{1}(G)$ be its group algebra, let $M(G)$ be its usual measure algebra, let $L^{1}(G)^{**}$ be the second dual of $L^{1}(G)$ with an Arens product, and let $LUC(G)^{*}$ be the conjugate of the space $LUC(G)$ of bounded, left uniformly continuous, complex-valued functions on $G$ with an Arens-type product. We find all the finite-dimensional left ideals of these algebras. We deduce that such ideals exist in $L^{1}(G)$ and $M(G)$ if and only if $G$ is compact, and in $L^{1}(G)^{**}$ (except those generated by right annihilators of $L^{1}(G)^{**}$) and $LUC(G)^{*}$ if and only if $G$ is amenable.